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Radiation pressure: why cos 2 ?

Astro 300B: Jan. 24, 2011 Optical Depth Eddington Luminosity Thermal radiation and Thermal Equilibrium. Radiation pressure: why cos 2 ?. Each photon of energy E=h n has momentum h n /c. Want the component of the momentum normal To direction defined by dA, this will be.

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Radiation pressure: why cos 2 ?

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  1. Astro 300B: Jan. 24, 2011Optical DepthEddington LuminosityThermal radiation and Thermal Equilibrium

  2. Radiation pressure: why cos2? Each photon of energy E=hn has momentum hn/c Want the component of the momentum normal To direction defined by dA, this will be Pressure = net momentum normal to dA/time/area or the net energy x cos q/c /time/area The net energy = flux in direction n, i.e.

  3. OPTICAL DEPTH It’s useful to rewrite the transfer equation in terms of the optical depth: or emergent incident >1: optically thick opaque, typical photon will be absorbed <1: optically thintransparent, typical photon will traverse the medium without being absorbed

  4. Source Function Define so that the Equation of Radiative Transfer is Which has solution Specific intensity decreases along path then If Specific intensity increases along path then If If >>1, so

  5. Mean Free Path is the average distance a photon travels before being absorbed The mean free path, Or in other words, the distance through the absorbing material corresponding to optical depth = 1 Recall where = absorption coefficient or so also hence #absorbers/Vol Cross-section for absorption

  6. Makes sense: decreases If N increases, decreases If σνincreases,

  7. Radiation Force: The Eddington Limitsee R&L Problem 1.4 and p. 15 • Photons carry momentum • When radiation is absorbed by a medium, it therefore exerts a force upon it Consider a source of radiation, with luminosity L (ergs/sec) And a piece of material a distance r from the source Each photon absorbed imparts momentum = E / c Specific flux = Fν ergs s-1 cm-2 Hz-1 Momentum flux = Fν / c momentum s-1 cm-2 Hz-1 Momentum imparted by absorbed photons = Where = absorption coefficient, cm-1

  8. Momentum /area /time /Hz /pathlength through absorber Now, area x pathlength = volume so Is momentum/time /Hz /volume But momentum /time = Force So, integrating over frequency, the Force/volume imparted by the absorbed photons is

  9. Likewise, in terms of the mass absorption coefficient, κ An important application of this concept is the Eddington Luminosity, or Eddington Limit This is the maximum luminosity an object can have before it ejects hydrogen by radiation pressure

  10. Eddington Luminosity c.f. Accretion onto a black hole r When does fgrav = fradiation? M,L Force per unit mass = force per unit mass due to gravity due to absorption of radiation F = radiation flux, integrated over frequency fradiation = L = luminosity of radiation, ergs/sec r = distance between blob and the source Κ = mass absorption coefficent

  11. fgravity = M = mass of the source So… Define Eddington Luminosity = the L at which f(gravity) = f(radiation)

  12. Maximum luminosity of a source of mass M Note: independent of r A “minimal” value for κ is the Thomson cross-section, For Thomson scattering of photons off of free electrons, assuming the gas is completely ionized and pure hydrogen Other sources of absorption opacity, if present, will contribute to larger κ, and therefore smaller L

  13. Thomson cross-section σT = 6.65 x 10-25 cm2 Independent of frequency (except at very high frequencies) Where mH= mass of hydrogen atom

  14. If M = M(Sun), then Ledd = 1.25 x 1038 erg/sec Compared to L(sun) = 3.9 x 1033 erg/sec

  15. Another example of a cross-section for absorption: Photoionization of Hydrogen from the ground state H atom + hν p + e- Only photons more energetic thanthreshold χ can ionize hydrogen, where χ = 13.6 eV 912 Å 1 Rydberg Lyman limit ν1 = 3.3x1015 sec-1 The cross-section for absorption is a function of frequency, where ν1=3.3x1015sec-1 More energetic photons are less likely to ionize hydrogen than photons at energies near the Lyman Limit Note: αν : photon-particle cross-sections; σν: particle-particle cross-sections

  16. Similarly, one can consider the ionization of He I  He II He II  He III Thresholds: Hydrogen hν = 13.6 eV 912 Å Helium I 24.6 eV 504 Å Helium II 54.4 eV 228 Å For HeIα(504 Å) = 7.4 x 10-18 cm2 declines like ν2 For HeIIα(228 Å) = 1.7 x 10-18 cm2 declines like ν3

  17. Thermal Radiation, and Thermodynamic Equilibrium

  18. Thermal radiation is radiation emitted by matter in thermodynamic equilibrium. When radiation is in thermal equilibrium, Iν is a universal function of frequency ν and temperature T – the Planck function, Bν. Blackbody Radiation: In a very optically thick media, recall the SOURCE FUNCTION So thermal radiation has And the equation of radiative transfer becomes

  19. THERMODYNAMIC EQUILIBRIUM When astronomers speak of thermodynamic equilibrium, they mean a lot more than dT/dt = 0, i.e. temperature is a constant. DETAILED BALANCE: rate of every reaction = rate of inverse reaction on a microprocess level • If DETAILED BALANCE holds, then one can describe • The radiation field by the Planck function • The ionization of atoms by the SAHA equation • The excitation of electroms in atoms by the Boltzman distribution • The velocity distribution of particles by the Maxwell-Boltzman distribution • ALL WITH THE SAME TEMPERATURE, T • When (1)-(4) are described with a single temperature, T, then the system is • said to be in THERMODYNAMIC EQUILIBRIUM.

  20. In thermodynamic equilibrium, the radiation and matter have the same temperature, i.e. there is a very high level of coupling between matter and radiation  Very high optical depth By contrast, a system can be in statistical equilibrium, or in a steady state, but not be in thermodynamic equilibrium. So it could be that measurable quantities are constant with time, but there are 4 different temperatures: T(ionization) given by the Saha equation T(excitation) given by the Boltzman equation T(radiation) given by the Planck Function T(kinetic) given by the Maxwell-Boltzmann distribution Where T(ionization) ≠ T(excitation) ≠ T(radiation) ≠ T(kinetic)

  21. LOCAL THERMODYNAMIC EQUILIBRIUM (LTE) If locally, T(ion) = T(exc) = T(rad) = T(kinetic) Then the system is in LOCAL THERMODYNAMIC EQUILIBRIUM, or LTE This can be a good approximation if the mean free path for particle-photon interactions << scale upon which T changes

  22. Example: H II Region (e.g. Orion Nebula, Eagle Nebula, etc) Ionized region of interstellar gas around a very hot star Radiation field is essentially a black-body at the temperature of the central Star, T~50,000 – 100,000 K However, the gas cools to Te ~ 10,000 K (Te = kinetic temperature of electrons) O star H II H I Q.: Is this room in thermodynamic equilibrium?

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